Subgroups — #30640959

Order 16256.

140 conjugacy classes of subgroups.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#22550256	C₂	1	1	yes	maximal, characteristic
2	#22549679	C₂	1	4	yes	maximal
3	C₄×C₁₂₇⋊C₁₆	C₂	1	1	yes	maximal, characteristic
4	#500×C₁₂₇	C₂	1	1	yes	maximal, characteristic
5	C₂×Dic₅₀₈	—	2	1	no	—
6	C₂×Dic₅₀₈	C₂²	1	2	yes	—
7	#16861209	—	2	1	no	—
8	C₂×C₁₂₇⋊C₁₆	C₂²	1	2	yes	—
9	C₄⋊Q₈×C₁₂₇	—	2	1	no	—
10	#107×C₁₂₇	—	2	1	no	—
11	#107×C₁₂₇	C₂²	1	2	yes	—
12	C₄×C₈×C₁₂₇	C₂²	1	1	yes	characteristic
13	C₂×Dic₂₅₄	—	2	2	no	—
14	Dic₅₀₈	—	2	4	no	—
15	Dic₅₀₈	—	4	2	no	—
16	C₄⋊Dic₁₂₇	—	4	1	no	—
17	C₄⋊Dic₁₂₇	—	4	1	no	—
18	C₁₂₇⋊C₁₆	—	2	2	no	—
19	C₂×Q₈×C₁₂₇	—	4	1	no	—
20	C₂×C₈×C₁₂₇	D₄	1	1	yes	characteristic
21	C₂×C₈×C₁₂₇	C₂³	1	1	yes	characteristic
22	C₄⋊C₄×C₁₂₇	—	2	2	no	—
23	C₄⋊C₄×C₁₂₇	—	4	1	no	—
24	C₄²×C₁₂₇	D₄	1	1	yes	characteristic
25	Dic₂₅₄	—	4	2	no	—
26	Dic₂₅₄	—	4	2	no	—
27	C₂×Dic₁₂₇	—	4	2	no	—
28	Q₈×C₁₂₇	—	8	2	no	—
29	C₂×C₄×C₁₂₇	—	4	2	no	—
30	C₂×C₄×C₁₂₇	D₈	1	1	yes	characteristic
31	C₂×C₄×C₁₂₇	D₈	1	1	yes	characteristic
32	C₂×C₄×C₁₂₇	C₂×D₄	1	1	yes	characteristic
33	C₈×C₁₂₇	SD₁₆	1	2	yes	—
34	C₈×C₁₂₇	#41	1	2	yes	—
35	C₂²×C₁₂₇	C₂×D₈	1	1	yes	characteristic
36	C₄×C₁₂₇	—	8	2	no	—
37	C₄×C₁₂₇	SD₃₂	1	2	yes	—
38	C₄×C₁₂₇	Q₃₂	1	2	yes	—
39	C₄×C₁₂₇	#124	1	1	yes	characteristic
40	C₄×C₁₂₇	C₂×SD₁₆	1	1	yes	characteristic
41	Dic₁₂₇	—	8	2	no	—
42	C₂×C₁₂₇	#486	1	1	yes	characteristic
43	C₂×C₁₂₇	C₂×SD₃₂	1	1	yes	characteristic
44	C₂×C₁₂₇	C₂×Q₃₂	1	1	yes	characteristic
45	#2241	—	127	1	no	maximal, Sylow
46	C₁₂₇	#2241	1	1	yes	characteristic, complemented, Sylow
47	#500	D₁₂₇	1	1	yes	characteristic
48	#494	—	127	1	no	—
49	#358	—	127	4	no	—
50	C₄×C₁₆	—	127	1	no	—
51	C₂×Q₁₆	—	127	2	no	—
52	C₂×Q₁₆	—	254	1	no	—
53	C₄⋊Q₈	—	2	1	no	—
54	C₄⋊Q₈	—	254	1	no	—
55	C₂×C₁₆	—	127	2	no	—
56	#107	—	2	1	no	—
57	#107	C₂×D₁₂₇	1	2	yes	—
58	C₄×C₈	C₂×D₁₂₇	1	1	yes	characteristic
59	C₂×Q₈	—	4	1	no	—
60	C₂×Q₈	—	254	2	no	—
61	Q₁₆	—	508	2	no	—
62	Q₁₆	—	254	4	no	—
63	C₂×C₈	C₂²⋊D₁₂₇	1	1	yes	characteristic
64	C₂×C₈	C₂²×D₁₂₇	1	1	yes	characteristic
65	C₄⋊C₄	—	2	2	no	—
66	C₄⋊C₄	—	4	1	no	—
67	C₄⋊C₄	—	508	1	no	—
68	C₄⋊C₄	—	508	1	no	—
69	C₄²	C₂²⋊D₁₂₇	1	1	yes	characteristic
70	C₁₆	—	254	2	no	—
71	Q₈	—	8	2	no	—
72	Q₈	—	508	2	no	—
73	Q₈	—	508	2	no	—
74	C₂×C₄	—	4	2	no	—
75	C₂×C₄	—	508	2	no	—
76	C₂×C₄	#14523233	1	1	yes	characteristic
77	C₂×C₄	#14523233	1	1	yes	characteristic
78	C₂×C₄	C₂×C₂²⋊D₁₂₇	1	1	yes	characteristic
79	C₈	#14523235	1	2	yes	—
80	C₈	#14523249	1	2	yes	—
81	C₂²	C₂×#14523233	1	1	yes	characteristic
82	C₄	—	8	2	no	—
83	C₄	—	1016	2	no	—
84	C₄	#16861153	1	2	yes	—
85	C₄	#16861155	1	2	yes	—
86	C₄	#16861198	1	1	yes	characteristic
87	C₄	C₂×#14523235	1	1	yes	characteristic
88	C₂	#22550224	1	1	yes	characteristic
89	C₂	C₂×#16861153	1	1	yes	characteristic
90	C₂	C₂×#16861155	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂×#14523233
commutator, cosocle	C₂×C₈×C₁₂₇	C₂³
Frattini	C₂×C₈	C₂²×D₁₂₇
Fitting	#500×C₁₂₇	C₂
socle, prime radical	C₂²×C₁₂₇	C₂×D₈
hypercenter	#500	D₁₂₇
hypocenter	C₁₂₇	#2241
LCS	G ▷ C₂×C₈×C₁₂₇ ▷ C₄×C₁₂₇ ▷ C₂×C₁₂₇ ▷ C₁₂₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #500

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
46. C₁₂₇	#2241	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#22550256	2
#22549679	2
C₄×C₁₂₇⋊C₁₆	2
#500×C₁₂₇	2
#2241	127